## Appendix: The multilateral TFP index

As detailed in Fukao et al. (2011), the TFP level of firm i in industry j in year t, $ TFP_{i,j,t} $ is defined in comparison with the TFP level of a hypothetical representative firm in the benchmark year $ t_{0} $ in industry j. In the EALC 2010 Database, the benchmark year $ t_{0} $ is set to the year 2000 and the firm-level TFP level is calculated as follows, using the multilateral TFP index method developed by Good et al. (1997):.

$$ \begin{aligned}&\begin{aligned}\\ &\mathrm{LN}\big(\mathrm{TFP}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)=\left\{\mathrm{LN}\big(\mathrm{Q}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{t}}\big)\right\}-\sum_{\mathrm{k}=1}^{\mathrm{n}}\big(\mathrm{S}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}+\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{t}}}\big)\left\{\mathrm{LN}\big(\mathrm{X}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{t}}\big)\right\}\\&\quad\mathrm{for}\mathrm{t}=\mathrm{t}_{0}\\ &\end{aligned}\\ &\\&\mathrm{LN}\big(\mathrm{TFP}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)=\left\{\mathrm{LN}\big(\mathrm{Q}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{t}}\big)\right\}-\frac{1}{2}\sum_{\mathrm{k}=1}^{\mathrm{n}}\big(\mathrm{S}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}+\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{t}}}\big)\left\{\mathrm{LN}\big(\mathrm{X}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{t}}\big)\right\}\\&\quad+\sum_{\mathrm{s}=\mathrm{t}_{0}+1}^{\mathrm{t}}\left\{\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{s}}\big)-\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{s}-1}\big)\right\}\\&\quad-\sum_{\mathrm{s}=\mathrm{t}_{0}+1}^{\mathrm{t}}\sum_{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{2}\big(\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{s}}}+\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{s}-1}}\big)\left\{\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{s}}\big)-\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{s}-1}\big)\right\}\\&\quad\mathrm{for}\mathrm{t}>\mathrm{t}_{0}\\ &\\&\mathrm{LN}\big(\mathrm{TFP}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)=\left\{\mathrm{LN}\big(\mathrm{Q}_{\mathrm{i},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{t}}\big)\right\}-\frac{1}{2}\sum_{\mathrm{k}=1}^{\mathrm{n}}\big(\mathrm{S}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}+\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{t}}}\big)\left\{\mathrm{LN}\big(\mathrm{X}_{\mathrm{i},\mathrm{k},\mathrm{j},\mathrm{t}}\big)-\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{t}}\big)\right\}\\&\quad-\sum_{\mathrm{s}=\mathrm{t}+1}^{\mathrm{t}_{0}}\left\{\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{s}}\big)-\overline{\mathrm{LN}}\big(\mathrm{Q}_{\mathrm{j},\mathrm{s}-1}\big)\right\}+\sum_{\mathrm{s}=\mathrm{t}+1}^{\mathrm{t}_{0}}\sum_{\mathrm{k}=1}^{\mathrm{n}}\frac{1}{2}\big(\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{s}}}+\overline{\mathrm{S}_{\mathrm{k},\mathrm{j},\mathrm{s}-1}}\big)\left\{\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{s}}\big)-\overline{\mathrm{LN}}\big(\mathrm{X}_{\mathrm{k},\mathrm{j},\mathrm{s}-1}\big)\right\}\\&\quad\mathrm{for}\mathrm{t}<\mathrm{t}_{0}\\ &\end{aligned}\\ \end{aligned} $$

where $ Q_{i,j,t} $ stands for the real output (real sales) of firm i (in industry j) in year t, $ X_{i,k,j,t} $ represents the real input of production factor k of firm i (in industry j) in year t, and $ S_{i,j,k,t} $ is the cost share of production factor k at firm i (in industry j) in year t. $ \overline{LN(Q_{j,t})} $ denotes the arithmetic average of the log value of the output, in year t, of all firms in industry j to which firm i belongs, while $ \overline{LN(X_{k,j,t})} $ stands for the arithmetic average of the log value of the input of production factor k, in year t, of all firms in industry j to which firm i belongs. Finally, $ \overline{S_{k,j,t}} $ is the arithmetic average of the cost share of the input of production factor k, in year t, of all firms in industry j to which firm i belongs.